Theorem wrddun

Description: Words in either of two alphabets are words in their union. (Contributed by Ender Ting, 24-Jul-2026)

	Ref	Expression
Assertion	wrddun	⊢ ( ( 𝐴 ∈ Word 𝐵 ∨ 𝐴 ∈ Word 𝐶 ) → 𝐴 ∈ Word ( 𝐵 ∪ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssun1	⊢ 𝐵 ⊆ ( 𝐵 ∪ 𝐶 )
2		sswrd	⊢ ( 𝐵 ⊆ ( 𝐵 ∪ 𝐶 ) → Word 𝐵 ⊆ Word ( 𝐵 ∪ 𝐶 ) )
3	1 2	ax-mp	⊢ Word 𝐵 ⊆ Word ( 𝐵 ∪ 𝐶 )
4	3	sseli	⊢ ( 𝐴 ∈ Word 𝐵 → 𝐴 ∈ Word ( 𝐵 ∪ 𝐶 ) )
5		ssun2	⊢ 𝐶 ⊆ ( 𝐵 ∪ 𝐶 )
6		sswrd	⊢ ( 𝐶 ⊆ ( 𝐵 ∪ 𝐶 ) → Word 𝐶 ⊆ Word ( 𝐵 ∪ 𝐶 ) )
7	5 6	ax-mp	⊢ Word 𝐶 ⊆ Word ( 𝐵 ∪ 𝐶 )
8	7	sseli	⊢ ( 𝐴 ∈ Word 𝐶 → 𝐴 ∈ Word ( 𝐵 ∪ 𝐶 ) )
9	4 8	jaoi	⊢ ( ( 𝐴 ∈ Word 𝐵 ∨ 𝐴 ∈ Word 𝐶 ) → 𝐴 ∈ Word ( 𝐵 ∪ 𝐶 ) )